Latitude, usually denoted by the Greek letter
phi (
φ) gives the
location of a place on
Earth (or other
planetary body) north or south of the
equator.
Lines of Latitude are the
imaginary horizontal lines shown running east-to-west (or west to
east) on maps (particularly so in the
Mercator projection) that run either
north or south of the equator.
Technically, latitude is an angular measurement in degrees (marked with °) ranging from 0° at
the equator (low latitude) to 90° at
the poles (90° N or +90° for the North Pole
and 90° S or −90° for the South Pole
). The latitude is approximately the angle
between straight up at the surface (the
zenith) and the sun at an
equinox. The
complementary angle of a latitude is
called the
colatitude.
Circles of latitude
All locations of a given latitude are collectively referred to as a
circle of latitude or
line of latitude or
parallel, because they are
coplanar, and all such
plane are
parallel to the
equator. Lines of latitude other than the Equator
are approximately
small circles on the
surface of the Earth; they are not
geodesics since the shortest route between two
points at the same latitude involves a path that bulges toward the
nearest pole, first moving farther away from and then back toward
the equator (see
great circle).
A specific latitude may then be combined with a specific
longitude to give a precise position on the
Earth's surface (see
satellite navigation
system).
Important named circles of latitude
Besides the equator, four other lines of latitude are named because
of the role they play in the geometrical relationship with the
Earth and the Sun:
Only at latitudes between the Tropics is it possible for the
sun to be at the
zenith.
Only north of the
Arctic Circle or
south of the
Antarctic Circle is
the
midnight sun possible.
The reason that these lines have the values that they do lies in
the
axial tilt of the Earth with respect
to the sun, which is
23° 26′
21.41″.
Note that the Arctic Circle and Tropic of Cancer are colatitudes,
since the sum of their angles is 90°—similarly for the Antarctic
Circle and Tropic of Capricorn.
Subdivisions
A degree is divided into 60
minutes.
One minute can be further divided into 60 seconds. An example of a
latitude specified in this way is 13°19'43″ N (for greater
precision, a decimal fraction can be added to the seconds). An
alternative representation uses only degrees and minutes, where the
seconds are expressed as a decimal fraction of minutes: the above
example would be expressed as 13°19.717' N. Degrees can also be
expressed singularly, with both the minutes and seconds
incorporated as a decimal number and rounded as desired (decimal
degree notation): 13.32861° N.
Sometimes, the north/south suffix is replaced
by a negative sign for south (−90° for the South Pole).
Effect of latitude
Average temperatures vary strongly
with latitude.
A region's latitude has a great effect on its
climate and
weather (see
Effect of sun angle
on climate). Latitude more loosely determines tendencies
in
polar auroras,
prevailing winds, and other physical
characteristics of geographic locations.
Researchers at Harvard
's Center for
International Development (CID) found in 2001 that only three
tropical economies — Hong Kong
, Singapore
, and Taiwan
— were
classified as high-income by the World
Bank, while all countries within regions zoned as temperate had either middle- or high-income
economies. The validity of the Harvard report may be
questioned because a different threshold is used for the tropical
regions and the World Bank list fails to include Qatar's, United
Arab Emirates', and Kuwait's economies. Further, countries such as
Brazil have far better incomes than much of the Former Soviet Union
and Iron Curtain states .
Elliptic parameters
Because most planets (including Earth) are
ellipsoids of
revolution, or
spheroids,
rather than
spheres, both the radius and the
length of arc varies with latitude. This variation requires the
introduction of elliptic parameters based on an ellipse's
angular
eccentricity, o\!\varepsilon\,\! (which equals
\arccos\left(\frac{b}{a}\right)\,\!, where a\;\! and b\;\! are the
equatorial radius (6378137.0 m for Earth) and the polar radius
(6356752.3142 m for Earth), respectively;
\sin^2(o\!\varepsilon)\,\! is the
first eccentricity squared,
{e^2}\,\!; and 2\sin^2\left(\frac{o\!\varepsilon}{2}\right)\;\! or
1-\cos(o\!\varepsilon)\,\! is the
flattening, {f}\,\!). Utilized in creating the
integrand for
curvature is the inverse of the
principal elliptic integrand, E'\,\!:
- :
n'(\phi)=\frac{1}{E'(\phi)}
=\frac{1}{\sqrt{1-(\sin(\phi)\sin(o\!\varepsilon))^2}};\,\!
- :\begin{align}
M(\phi)&=a\cdot\cos^2(o\!\varepsilon)n'^3(\phi)
=\frac{(ab)^2}{\Big((a\cos(\phi))^2+(b\sin(\phi))^2\Big)^{3/2}};\\
N(\phi)&=a{\cdot}n'(\phi)
=\frac{a^2}{\sqrt{(a\cos(\phi))^2+(b\sin(\phi))^2}}.\end{align}\,\!
Degree length
On Earth, the length of an
arcdegree
of north–south latitude difference, \scriptstyle{\Delta\phi}\,\!,
is about 60
nautical miles, 111
kilometres or 69
statute miles at any latitude. The length of an
arcdegree of east-west longitude difference,
\scriptstyle{\cos(\phi)\Delta\lambda}\,\!, is about the same at the
equator as the north-south, reducing to zero at the poles.
In the case of a spheroid, a
meridian and its anti-meridian form an
ellipse, from which an exact expression for
the length of an arcdegree of latitude difference is:
- :\frac{\pi}{180^\circ}M(\phi);\,\!
This radius of arc (or "arcradius") is in the plane of a meridian,
and is known as the
meridional radius of curvature,
M\,\!.
Similarly, an exact expression for the length of an arcdegree of
longitude difference is:
- :\frac{\pi}{180^\circ}\cos(\phi)N(\phi);\,\!
The arcradius contained here is in the plane of the
prime vertical, the east-west plane
perpendicular (or "
normal") to both
the plane of the meridian and the plane tangent to the surface of
the ellipsoid, and is known as the
normal radius of
curvature, N\,\!.
Along the equator (east-west), N\;\! equals the equatorial radius.
The radius of curvature at a
right angle
to the equator (north-south), M\;\!, is 43 km shorter, hence
the length of an arcdegree of latitude difference at the equator is
about 1 km less than the length of an arcdegree of longitude
difference at the equator. The radii of curvature are equal at the
poles where they are about 64 km greater than the north-south
equatorial radius of curvature
because the polar radius is
21 km less than the equatorial radius. The shorter polar radii
indicate that the northern and southern hemispheres are flatter,
making their radii of curvature longer. This flattening also
'pinches' the north-south equatorial radius of curvature, making it
43 km less than the equatorial radius. Both radii of curvature
are perpendicular to the plane tangent to the surface of the
ellipsoid at all latitudes, directed toward a point on the polar
axis in the opposite hemisphere (except at the equator where both
point toward Earth's center). The east-west radius of curvature
reaches the axis, whereas the north-south radius of curvature is
shorter at all latitudes except the poles.
The
WGS84 ellipsoid, used by all
GPS devices, uses an equatorial
radius of 6378137.0 m and an inverse flattening, (1/f), of
298.257223563, hence its polar radius is 6356752.3142 m and its
first eccentricity squared is 0.00669437999014. The more recent but
little used
IERS 2003 ellipsoid provides
equatorial and polar radii of 6378136.6 and 6356751.9 m,
respectively, and an inverse flattening of 298.25642. Lengths of
degrees on the WGS84 and IERS 2003 ellipsoids are the same when
rounded to six
significant digits.
An appropriate calculator for any latitude is provided by the U.S.
government's
National
Geospatial-Intelligence Agency (NGA).
| Latitude |
N-S radius
of curvature
M\;\! |
Surface distance
per 1° change
in latitude |
|
E-W radius
of curvature
N\;\! |
Surface distance
per 1° change
in longitude |
| 0° |
6335.44 km |
110.574 km |
|
6378.14 km |
111.320 km |
| 15° |
6339.70 km |
110.649 km |
|
6379.57 km |
107.551 km |
| 30° |
6351.38 km |
110.852 km |
|
6383.48 km |
96.486 km |
| 45° |
6367.38 km |
111.132 km |
|
6388.84 km |
78.847 km |
| 60° |
6383.45 km |
111.412 km |
|
6394.21 km |
55.800 km |
| 75° |
6395.26 km |
111.618 km |
|
6398.15 km |
28.902 km |
| 90° |
6399.59 km |
111.694 km |
|
6399.59 km |
0.000 km |
Types of latitude
With a spheroid that is slightly flattened by its rotation,
cartographers refer to a variety of auxiliary latitudes to
precisely adapt spherical projections according to their
purpose.
For planets other than Earth, such as
Mars,
geographic and geocentric latitude are called "planetographic" and
"planetocentric" latitude, respectively. Most maps of Mars since
2002 use planetocentric coordinates.
Common "latitude"
In common usage, "latitude" refers to
geodetic or
geographic
latitude \phi\,\! and is the angle between the
equatorial plane and a line that is
normal to the
reference ellipsoid, which approximates
the shape of Earth to account for flattening of the poles and
bulging of the equator. This value usually differs from the
geocentric
latitude.
The expressions following assume elliptical polar sections and that
all sections parallel to the equatorial plane are circular.
Geographic latitude (with longitude) then provides a
Gauss map. As defined earlier in this article,
o\!\varepsilon\,\! is the
angular
eccentricity of a meridian.
Reduced latitude
- On a spheroid, lines of reduced or
parametric latitude, \beta\,\!, form circles whose
radii are the same as the radii of circles formed by the
corresponding lines of latitude on a sphere with radius equal to
the equatorial radius of the spheroid.
- :\beta=\arctan\Big(\cos(o\!\varepsilon)\tan(\phi)\Big) =
\arctan\Bigg(\frac{b}{a}\tan(\phi)\Bigg);\,\!
Authalic latitude
- Authalic latitude, \xi\,\!, gives an
area-preserving transform to the sphere.
-
:\widehat{S}^2(\phi)=\frac{1}{2}b^2\left(\sin(\phi)n'^2(\phi)+\frac{\ln\bigg(n'(\phi)\Big(1+\sin(\phi)\sin(o\!\varepsilon)\Big)\bigg)}{\sin(o\!\varepsilon)}\right);\,\!
-
:\begin{align}\xi&=\arcsin\!\left(\frac{\widehat{S}^2(\phi)}{\widehat{S}^2(90^\circ)}\right),\\
&=\arcsin\!\left(\frac{\sin(\phi)\sin(o\!\varepsilon)n'^2(\phi)+\ln\Big(n'(\phi)\big(1+\sin(\phi)\sin(o\!\varepsilon)\big)\Big)}{\sin(o\!\varepsilon)\sec^2(o\!\varepsilon)+\ln\Big(\sec(o\!\varepsilon)\big(1+\sin(o\!\varepsilon)\big)\Big)}\right);\end{align}\,\!
Rectifying latitude
- Rectifying latitude, \mu\,\!, is the surface
distance from the equator, scaled so the pole is 90°, but involves
elliptic integration:
-
:: \mu=\frac{\;\int_{0}^\phi\;M(\theta)\,d\theta}{\frac{2}{\pi}\int_{0}^{90^\circ}M(\phi)\,d\phi}
=\frac{\pi}{2}\cdot\frac{\;\int_{0}^\phi\;n'^3(\theta)\,d\theta}{\int_{0}^{90^\circ}n'^3(\phi)\,d\phi};\,\!
Conformal latitude
- Conformal latitude, \chi\,\!, gives an
angle-preserving (conformal) transform to
the sphere.
-
:\chi=2\cdot\arctan\left(\sqrt{\frac{1+\sin(\phi)}{1-\sin(\phi)}\cdot\left(\frac{1-\sin(\phi)\sin(o\!\varepsilon)}{1+\sin(\phi)\sin(o\!\varepsilon)}\right)^{\!\!\sin(o\!\varepsilon)}}^{\color{white}|}\;\right)-\frac{\pi}{2};\;\!
Geocentric latitude
- The geocentric latitude, \psi\,\!, is the
angle between the equatorial plane and a line from the center of
Earth.
- :\psi=\arctan\Big(\cos^2(o\!\varepsilon)\tan(\phi)\Big) =
\arctan\Big((b/a)^2\tan(\phi)\Big).\;\!
- It is the size of the central
angle between the equator and the point of interest, as
measured along a meridian. This
value usually differs from the geographic latitude, as so:
Illustration of geographic and
geocentric latitudes.
Astronomical latitude
A more obscure measure of latitude is the
astronomical
latitude, which is the angle between the equatorial plane
and the
normal to the
geoid (ie a plumb line). It originated as the angle
between horizon and pole star. It differs from the geodetic
latitude only slightly, due to the slight deviations of the geoid
from the reference ellipsoid.
Astronomical latitude is not to be confused with
declination, the coordinate
astronomers use to describe the locations of
stars north/south of the
celestial
equator (see
equatorial
coordinates), nor with
ecliptic
latitude, the coordinate that astronomers use to describe the
locations of stars north/south of the
ecliptic (see
ecliptic coordinates).
Palaeolatitude
Continents move over time, due to
continental drift, taking whatever fossils
and other features of interest they may have with them.
Particularly when discussing fossils, it's often more useful to
know where the fossil was when it was laid down, than where it is
when it was dug up: this is called the
palæolatitude of
the fossil. The Palæolatitude can be constrained by
palæomagnetic data. If tiny magnetisable
grains are present when the rock is being formed, these will align
themselves with Earth's magnetic field like compass needles. A
magnetometer can deduce the orientation
of these grains by subjecting a sample to a magnetic field, and the
magnetic declination of the
grains can be used to infer the latitude of deposition.
Comparison of selected types
The following plot shows the differences between the types of
latitude. The data used are found in the table following the plot.
Please note that the values in the table are in minutes, not
degrees, and the plot reflects this as well. Also observe that the
conformal symbols are hidden behind the geocentric due to being
very close in value. Finally it is important to mention also that
these differences don't mean that the use of one specific latitude
will necessarily cause more distortions than the other (the real
fact is that each latitude type is optimized for achieving a
different goal).
- {| class="wikitable"
Corrections for altitude
Line
IHis normal to the spheroid representing the Earth
(colored orange) at point
H.
The angle it forms with the equator (represented by line
CA) corresponds to the point's geodetic latitude.
When converting from geodetic ("common") latitude to other types of
latitude, corrections must be made for altitude for systems which
do not measure the angle from the
normalof the
spheroid. For example, in the figure at right,
point
H(located on the surface of the spheroid) and point
H''' (located at some greater elevation) have different
''geocentric'' latitudes (angles ''β'' and ''γ'' respectively),
even though they share the same ''geodetic'' latitude (angle
''α''). Note that the flatness of the spheroid and elevation of
point ''H'in the image is significantly greater than what is
found on the Earth, exaggerating the errors inherent in such
calculations if left uncorrected. Note also that the
reference ellipsoidused in the geodetic
system is itself just an approximation of the true
geoid, and therefore introduces its own errors, though
the differences are less severe. (See
Astronomical latitude, above.)
See also
Footnotes
- Location, Location, Location. The relationship of
climate to, and the effect of disease and agricultural productivity
on, the economic success of a city or region.
- The Math Forum
- John P. Snyder, Map
Projections: A Working Manual (1987) 24-25
- NIMA TR8350.2 page 3-1.
- IERS Conventions (2003) (Chp. 1, page 12)
- Length of degree calculator - National
Geospatial-Intelligence Agency
External links
- Free GeoCoder
- GEONets Names Server, access to the National
Geospatial-Intelligence Agency's (NGA) database of foreign
geographic feature names.
- Look-up Latitude and Longitude
- Resources for determining your latitude and
longitude
- Convert decimal degrees into degrees, minutes, seconds
- Info about decimal to sexagesimal
conversion
- Convert decimal degrees into degrees, minutes,
seconds
- Latitude and longitude converter – Convert latitude
and longitude from degree, decimal form to degree, minutes, seconds
form and vice versa. Also included a farthest point and a distance
calculator.
- Worldwide Index
- Tageo.com – contains 2,700,000 coordinates of places
including US towns
- for each city it gives the satellite map location, country,
province, coordinates (dd,dms), variant names and nearby
places.
| Approximate difference from geographic latitude
("Lat") |
|
Lat
\phi\,\! |
Reduced
\phi-\beta\,\! |
Authalic
\phi-\xi\,\! |
Rectifying
\phi-\mu\,\! |
Conformal
\phi-\chi\,\! |
Geocentric
\phi-\psi\,\! |
|
| 0° |
0.00′ |
0.00′ |
0.00′ |
0.00′ |
0.00′ |
|
| 5° |
1.01′ |
1.35′ |
1.52′ |
2.02′ |
2.02′ |
|
| 10° |
1.99′ |
2.66′ |
2.99′ |
3.98′ |
3.98′ |
|
| 15° |
2.91′ |
3.89′ |
4.37′ |
5.82′ |
5.82′ |
|
| 20° |
3.75′ |
5.00′ |
5.62′ |
7.48′ |
7.48′ |
|
| 25° |
4.47′ |
5.96′ |
6.70′ |
8.92′ |
8.92′ |
|
| 30° |
5.05′ |
6.73′ |
7.57′ |
10.09′ |
10.09′ |
|
| 35° |
5.48′ |
7.31′ |
8.22′ |
10.95′ |
10.96′ |
|
| 40° |
5.75′ |
7.66′ |
8.62′ |
11.48′ |
11.49′ |
|
| 45° |
5.84′ |
7.78′ |
8.76′ |
11.67′ |
11.67′ |
|
| 50° |
5.75′ |
7.67′ |
8.63′ |
11.50′ |
11.50′ |
|
| 55° |
5.49′ |
7.32′ |
8.23′ |
10.97′ |
10.98′ |
|
| 60° |
5.06′ |
6.75′ |
7.59′ |
10.12′ |
10.13′ |
|
| 65° |
4.48′ |
5.97′ |
6.72′ |
8.95′ |
8.96′ |
|
| 70° |
3.76′ |
5.01′ |
5.64′ |
7.52′ |
7.52′ |
|
| 75° |
2.92′ |
3.90′ |
4.39′ |
5.85′ |
5.85′ |
|
| 80° |
2.00′ |
2.67′ |
3.00′ |
4.00′ |
4.01′ |
|
| 85° |
1.02′ |
1.35′ |
1.52′ |
2.03′ |
2.03′ |
|
| 90° |
0.00′ |
0.00′ |
0.00′ |
0.00′ |
0.00′ |
: 23° 26′ 21″ N
: 23° 26′ 21″ S
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